Consider the function f(x)=ebx on the domain AxA. Assume b is a real, positive constant (a) Express f(x) as a complex Fourier series. Comment on the behaviour of the Fourier coefficients cn in the limit n. (b) Show that f can also be written as a real Fourier series, that is, an expansion in terms of sine and cosine functions. f(x)=a0+k=1akcos(2kx/L)+bksin(2kx/L) (c) Compare f(x) with the truncated Fourier series and show that the series approaches the true function as the number of terms is increased. You'll find the real series easier to work with than the complex one. Plot f(x) and the series on the same panel. On a separate panel plot the residuals to show that they decrease as the number of terms is increased. (d) Consider g(x)=df/dx. Write down and sketch this function and note the behaviour around x=0. Derive the Fourier series for g by differentiating the result from part (b). Comment on the behaviour of the Fourier coefficients in the limit of large n for this case. Once again show graphically the convergence of the Fourier series to to g(x) as was done for f(x). Consider the function f(x)=ebx on the domain AxA. Assume b is a real, positive constant (a) Express f(x) as a complex Fourier series. Comment on the behaviour of the Fourier coefficients cn in the limit n. (b) Show that f can also be written as a real Fourier series, that is, an expansion in terms of sine and cosine functions. f(x)=a0+k=1akcos(2kx/L)+bksin(2kx/L) (c) Compare f(x) with the truncated Fourier series and show that the series approaches the true function as the number of terms is increased. You'll find the real series easier to work with than the complex one. Plot f(x) and the series on the same panel. On a separate panel plot the residuals to show that they decrease as the number of terms is increased. (d) Consider g(x)=df/dx. Write down and sketch this function and note the behaviour around x=0. Derive the Fourier series for g by differentiating the result from part (b). Comment on the behaviour of the Fourier coefficients in the limit of large n for this case. Once again show graphically the convergence of the Fourier series to to g(x) as was done for f(x)